England made 229 in their 32 overs after a storm halted play for three hours and Sri Lanka's target was subject to a Duckworth Lewis amendment.
England captain Alastair Cook, in his first game in permanent charge of England's one-day side, was delighted with the performance.
"It was an outstanding performance from all the bowlers with Jimmy up front," he said.
"Full credit to him, he deserved the wickets and he was begging me for one more over so he could get five.
"But we batted very well and the way we handled that rain break was very special." His Sri Lankan counterpart Tillakaratne Dilshan wasn't too disappointed by the defeat.
"The short game can go either way. They batted very well and we had to take a chance but it didn't work.
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Anderson's opening spell of 3 for 14 from four overs knocked the stuffing out of the Sri Lankan top order and their